Firstly, I should point out that most sexual crime is committed by someone the victim knows - 'stranger danger' is less of a fear than 'partner danger'... or in many circumstances 'pimp danger'.
Rape and sexual assault are generally not about sex, but have more to do with the assertion of power. Given this, laws providing greater access to sex (assuming that you have the economic power necessary to access that service) are highly unlikely to directly reduce sexual crime.
Laws against prostitution have many other negative social effects, but I don't think there is any direct evidence that they reduce most rape and sexual assault.
One possible exception to this is when a sexual assault/rape victim is a prostitute and prostitution is illegal. This provides potential encouragement for an assailant, who may choose their target in the knowledge that their victim may feel unable to report the crime - in the same way that a drug dealer is unlikely to report the theft of their illicit drugs. Crimes against prostitutes in areas where prostitution is illegal are higher than crimes against prostitutes where prostitution is lawful. It is not possible to simply state that this means those crimes are higher overall - it may be that where prostitution is legal such crimes are committed against other victims (again, seeking out those who are most vulnerable).
There is also the meme of the law enforcement officer who demands their 'share of the business' in order to 'turn a blind eye'. While it is impossible to say whether/how commonly this occurs, it would not be so likely in places where prostitution is legal.
While I support legal prostitution, this is based on the belief that consenting adults should be allowed to do what they want as long as there are no victims. While prostitution is illegal, the prostitute is a victim every day. I cannot say that sexual crimes would fall with legalisation of prostitution - but I can say that the lives of prostitutes are likely to be safer if they are legally recognised.
Edit (13 November 2017): there have been requests for academic sources, to which I feel I should respond.
There are papers saying that legal prostitution reduces rape. This article discusses UCLA research (published here) showing that lawful prostitution resulted in fewer rapes and lower rates of sexually transmitted disease. It is disappointing that the law in Rhode Island was subsequently changed to re-criminalise prostitution. This discussion paper (PDF) (a link to the final in this article about it was broken) also indicates that legalising prostitution - in The Netherlands - reduces rape. Unfortunately, it is extremely difficult to extrapolate from 'rape of prostitutes' to 'total rape' - especially when many sexual assaults are unreported.
It is on this basis that I have stated that 'there is no clear evidence that legalising prostitution reduces sexual crime'. It is simply too difficult to draw a straight line from point A to point B. One of the articles to which I have linked states about the research:
Without precise data on the victims of sexual violence, it is not
possible to determine exactly how the number of rapes and cases of
sexual abuse fall in the population at large. Some victims are sex
workers. But the authors believe the tippelzones lead “to a decrease
in sexual violence on women more generally by providing an anonymous,
appealing and easily accessible outlet for sex to otherwise violent
The authors cannot go beyond 'belief' in seeking to evaluate an overall effect on the population.
There are many more studies than the two to which I have referred, but they all have the same fundamental problem in trying to make the leap from 'this is what happens to prostitutes' to 'this is what happens in the general population'. Under-reporting and non-reporting of sexual assault means that it is not possible to give an absolute answer.
As a final part of this edit, I will provide two more references that are relevant. Firstly, an article titled Physical Abuse of Prostitutes is Common. The title speaks for itself, while the article focuses upon prostitutes who are 'working illegally'. The final link is to an Australian study, similarly looking at sexual assault of sex workers (PDF). While it is 40 pages, this is worth a read as it constitutes an in-depth examination of the issue in several jurisdictions.
I hope this additional information is useful. It does at least correct one part of my original comment. In paragraph 4 I identified sexual assault of prostitutes where their activities were illegal was a 'possible' exception. Strike the word 'possible' - it is clear from these studies that sex workers are more likely to be assaulted when they are operating illegally.